Integrand size = 16, antiderivative size = 150 \[ \int x^{5/2} (2-b x)^{5/2} \, dx=-\frac {5 \sqrt {x} \sqrt {2-b x}}{16 b^3}-\frac {5 x^{3/2} \sqrt {2-b x}}{48 b^2}-\frac {x^{5/2} \sqrt {2-b x}}{24 b}+\frac {1}{8} x^{7/2} \sqrt {2-b x}+\frac {1}{6} x^{7/2} (2-b x)^{3/2}+\frac {1}{6} x^{7/2} (2-b x)^{5/2}+\frac {5 \arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{8 b^{7/2}} \]
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Time = 0.03 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {52, 56, 222} \[ \int x^{5/2} (2-b x)^{5/2} \, dx=\frac {5 \arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{8 b^{7/2}}-\frac {5 \sqrt {x} \sqrt {2-b x}}{16 b^3}-\frac {5 x^{3/2} \sqrt {2-b x}}{48 b^2}+\frac {1}{6} x^{7/2} (2-b x)^{5/2}+\frac {1}{6} x^{7/2} (2-b x)^{3/2}+\frac {1}{8} x^{7/2} \sqrt {2-b x}-\frac {x^{5/2} \sqrt {2-b x}}{24 b} \]
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Rule 52
Rule 56
Rule 222
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^{7/2} (2-b x)^{5/2}+\frac {5}{6} \int x^{5/2} (2-b x)^{3/2} \, dx \\ & = \frac {1}{6} x^{7/2} (2-b x)^{3/2}+\frac {1}{6} x^{7/2} (2-b x)^{5/2}+\frac {1}{2} \int x^{5/2} \sqrt {2-b x} \, dx \\ & = \frac {1}{8} x^{7/2} \sqrt {2-b x}+\frac {1}{6} x^{7/2} (2-b x)^{3/2}+\frac {1}{6} x^{7/2} (2-b x)^{5/2}+\frac {1}{8} \int \frac {x^{5/2}}{\sqrt {2-b x}} \, dx \\ & = -\frac {x^{5/2} \sqrt {2-b x}}{24 b}+\frac {1}{8} x^{7/2} \sqrt {2-b x}+\frac {1}{6} x^{7/2} (2-b x)^{3/2}+\frac {1}{6} x^{7/2} (2-b x)^{5/2}+\frac {5 \int \frac {x^{3/2}}{\sqrt {2-b x}} \, dx}{24 b} \\ & = -\frac {5 x^{3/2} \sqrt {2-b x}}{48 b^2}-\frac {x^{5/2} \sqrt {2-b x}}{24 b}+\frac {1}{8} x^{7/2} \sqrt {2-b x}+\frac {1}{6} x^{7/2} (2-b x)^{3/2}+\frac {1}{6} x^{7/2} (2-b x)^{5/2}+\frac {5 \int \frac {\sqrt {x}}{\sqrt {2-b x}} \, dx}{16 b^2} \\ & = -\frac {5 \sqrt {x} \sqrt {2-b x}}{16 b^3}-\frac {5 x^{3/2} \sqrt {2-b x}}{48 b^2}-\frac {x^{5/2} \sqrt {2-b x}}{24 b}+\frac {1}{8} x^{7/2} \sqrt {2-b x}+\frac {1}{6} x^{7/2} (2-b x)^{3/2}+\frac {1}{6} x^{7/2} (2-b x)^{5/2}+\frac {5 \int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx}{16 b^3} \\ & = -\frac {5 \sqrt {x} \sqrt {2-b x}}{16 b^3}-\frac {5 x^{3/2} \sqrt {2-b x}}{48 b^2}-\frac {x^{5/2} \sqrt {2-b x}}{24 b}+\frac {1}{8} x^{7/2} \sqrt {2-b x}+\frac {1}{6} x^{7/2} (2-b x)^{3/2}+\frac {1}{6} x^{7/2} (2-b x)^{5/2}+\frac {5 \text {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )}{8 b^3} \\ & = -\frac {5 \sqrt {x} \sqrt {2-b x}}{16 b^3}-\frac {5 x^{3/2} \sqrt {2-b x}}{48 b^2}-\frac {x^{5/2} \sqrt {2-b x}}{24 b}+\frac {1}{8} x^{7/2} \sqrt {2-b x}+\frac {1}{6} x^{7/2} (2-b x)^{3/2}+\frac {1}{6} x^{7/2} (2-b x)^{5/2}+\frac {5 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{8 b^{7/2}} \\ \end{align*}
Time = 0.48 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.68 \[ \int x^{5/2} (2-b x)^{5/2} \, dx=\frac {\sqrt {x} \sqrt {2-b x} \left (-15-5 b x-2 b^2 x^2+54 b^3 x^3-40 b^4 x^4+8 b^5 x^5\right )}{48 b^3}-\frac {5 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}-\sqrt {2-b x}}\right )}{4 b^{7/2}} \]
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Time = 0.10 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.70
method | result | size |
meijerg | \(\frac {\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \left (-b \right )^{\frac {7}{2}} \left (-56 b^{5} x^{5}+280 b^{4} x^{4}-378 b^{3} x^{3}+14 b^{2} x^{2}+35 b x +105\right ) \sqrt {-\frac {b x}{2}+1}}{336 b^{3}}-\frac {5 \sqrt {\pi }\, \left (-b \right )^{\frac {7}{2}} \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{8 b^{\frac {7}{2}}}}{\left (-b \right )^{\frac {5}{2}} \sqrt {\pi }\, b}\) | \(105\) |
risch | \(-\frac {\left (8 b^{5} x^{5}-40 b^{4} x^{4}+54 b^{3} x^{3}-2 b^{2} x^{2}-5 b x -15\right ) \sqrt {x}\, \left (b x -2\right ) \sqrt {\left (-b x +2\right ) x}}{48 b^{3} \sqrt {-x \left (b x -2\right )}\, \sqrt {-b x +2}}+\frac {5 \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-b \,x^{2}+2 x}}\right ) \sqrt {\left (-b x +2\right ) x}}{16 b^{\frac {7}{2}} \sqrt {x}\, \sqrt {-b x +2}}\) | \(131\) |
default | \(-\frac {x^{\frac {5}{2}} \left (-b x +2\right )^{\frac {7}{2}}}{6 b}+\frac {-\frac {x^{\frac {3}{2}} \left (-b x +2\right )^{\frac {7}{2}}}{6 b}+\frac {5 \left (-\frac {3 \sqrt {x}\, \left (-b x +2\right )^{\frac {7}{2}}}{20 b}+\frac {3 \left (\frac {\left (-b x +2\right )^{\frac {5}{2}} \sqrt {x}}{3}+\frac {5 \left (-b x +2\right )^{\frac {3}{2}} \sqrt {x}}{6}+\frac {5 \sqrt {x}\, \sqrt {-b x +2}}{2}+\frac {5 \sqrt {\left (-b x +2\right ) x}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-b \,x^{2}+2 x}}\right )}{2 \sqrt {-b x +2}\, \sqrt {x}\, \sqrt {b}}\right )}{20 b}\right )}{6 b}}{b}\) | \(157\) |
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Time = 0.24 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.15 \[ \int x^{5/2} (2-b x)^{5/2} \, dx=\left [\frac {{\left (8 \, b^{6} x^{5} - 40 \, b^{5} x^{4} + 54 \, b^{4} x^{3} - 2 \, b^{3} x^{2} - 5 \, b^{2} x - 15 \, b\right )} \sqrt {-b x + 2} \sqrt {x} - 15 \, \sqrt {-b} \log \left (-b x + \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right )}{48 \, b^{4}}, \frac {{\left (8 \, b^{6} x^{5} - 40 \, b^{5} x^{4} + 54 \, b^{4} x^{3} - 2 \, b^{3} x^{2} - 5 \, b^{2} x - 15 \, b\right )} \sqrt {-b x + 2} \sqrt {x} - 30 \, \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{48 \, b^{4}}\right ] \]
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Timed out. \[ \int x^{5/2} (2-b x)^{5/2} \, dx=\text {Timed out} \]
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Time = 0.31 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.39 \[ \int x^{5/2} (2-b x)^{5/2} \, dx=\frac {\frac {15 \, \sqrt {-b x + 2} b^{5}}{\sqrt {x}} + \frac {85 \, {\left (-b x + 2\right )}^{\frac {3}{2}} b^{4}}{x^{\frac {3}{2}}} + \frac {198 \, {\left (-b x + 2\right )}^{\frac {5}{2}} b^{3}}{x^{\frac {5}{2}}} - \frac {198 \, {\left (-b x + 2\right )}^{\frac {7}{2}} b^{2}}{x^{\frac {7}{2}}} - \frac {85 \, {\left (-b x + 2\right )}^{\frac {9}{2}} b}{x^{\frac {9}{2}}} - \frac {15 \, {\left (-b x + 2\right )}^{\frac {11}{2}}}{x^{\frac {11}{2}}}}{24 \, {\left (b^{9} - \frac {6 \, {\left (b x - 2\right )} b^{8}}{x} + \frac {15 \, {\left (b x - 2\right )}^{2} b^{7}}{x^{2}} - \frac {20 \, {\left (b x - 2\right )}^{3} b^{6}}{x^{3}} + \frac {15 \, {\left (b x - 2\right )}^{4} b^{5}}{x^{4}} - \frac {6 \, {\left (b x - 2\right )}^{5} b^{4}}{x^{5}} + \frac {{\left (b x - 2\right )}^{6} b^{3}}{x^{6}}\right )}} - \frac {5 \, \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{8 \, b^{\frac {7}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 472 vs. \(2 (105) = 210\).
Time = 22.27 (sec) , antiderivative size = 472, normalized size of antiderivative = 3.15 \[ \int x^{5/2} (2-b x)^{5/2} \, dx=\frac {{\left ({\left ({\left (2 \, {\left ({\left (b x - 2\right )} {\left (4 \, {\left (b x - 2\right )} {\left (\frac {5 \, {\left (b x - 2\right )}}{b^{5}} + \frac {61}{b^{5}}\right )} + \frac {1251}{b^{5}}\right )} + \frac {3481}{b^{5}}\right )} {\left (b x - 2\right )} + \frac {11395}{b^{5}}\right )} {\left (b x - 2\right )} + \frac {11895}{b^{5}}\right )} \sqrt {{\left (b x - 2\right )} b + 2 \, b} \sqrt {-b x + 2} - \frac {6930 \, \log \left ({\left | -\sqrt {-b x + 2} \sqrt {-b} + \sqrt {{\left (b x - 2\right )} b + 2 \, b} \right |}\right )}{\sqrt {-b} b^{4}}\right )} b {\left | b \right |} - 36 \, {\left ({\left ({\left (2 \, {\left (b x - 2\right )} {\left ({\left (b x - 2\right )} {\left (\frac {4 \, {\left (b x - 2\right )}}{b^{4}} + \frac {41}{b^{4}}\right )} + \frac {171}{b^{4}}\right )} + \frac {745}{b^{4}}\right )} {\left (b x - 2\right )} + \frac {965}{b^{4}}\right )} \sqrt {{\left (b x - 2\right )} b + 2 \, b} \sqrt {-b x + 2} - \frac {630 \, \log \left ({\left | -\sqrt {-b x + 2} \sqrt {-b} + \sqrt {{\left (b x - 2\right )} b + 2 \, b} \right |}\right )}{\sqrt {-b} b^{3}}\right )} {\left | b \right |} + \frac {120 \, {\left ({\left ({\left (b x - 2\right )} {\left (2 \, {\left (b x - 2\right )} {\left (\frac {3 \, {\left (b x - 2\right )}}{b^{3}} + \frac {25}{b^{3}}\right )} + \frac {163}{b^{3}}\right )} + \frac {279}{b^{3}}\right )} \sqrt {{\left (b x - 2\right )} b + 2 \, b} \sqrt {-b x + 2} - \frac {210 \, \log \left ({\left | -\sqrt {-b x + 2} \sqrt {-b} + \sqrt {{\left (b x - 2\right )} b + 2 \, b} \right |}\right )}{\sqrt {-b} b^{2}}\right )} {\left | b \right |}}{b} - \frac {320 \, {\left (\sqrt {{\left (b x - 2\right )} b + 2 \, b} \sqrt {-b x + 2} {\left ({\left (b x - 2\right )} {\left (\frac {2 \, {\left (b x - 2\right )}}{b^{2}} + \frac {13}{b^{2}}\right )} + \frac {33}{b^{2}}\right )} - \frac {30 \, \log \left ({\left | -\sqrt {-b x + 2} \sqrt {-b} + \sqrt {{\left (b x - 2\right )} b + 2 \, b} \right |}\right )}{\sqrt {-b} b}\right )} {\left | b \right |}}{b^{2}}}{240 \, b} \]
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Timed out. \[ \int x^{5/2} (2-b x)^{5/2} \, dx=\int x^{5/2}\,{\left (2-b\,x\right )}^{5/2} \,d x \]
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